Properties of φ-δ-primary and 2-absorbing δ-primary ideals of commutative rings

Let R be a commutative ring with unity 1 not equal 0 and let delta be an ideal expansion. In the first part of this paper, we extend the concept of delta-primary ideals to phi-delta-primary ideals, where phi is an ideal reduction and delta is an ideal expansion. We introduce some of the ideal expansion delta and define phi-delta-primary ideals, where phi is an ideal reduction. Also, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized phi-delta-primary ideals with respect to such an ideal expansion delta. In the second part of this paper we investigate 2-absorbing delta-primary ideals which unify 2-absorbing ideals and 2-absorbing primary ideals, where delta is an ideal expansion. A number of results in the two parts are given.